Let $F_2$ be the free group of rank 2, and let $G$ be a finite 2-generated group. To any such $G$, we may associate the set of surjections $$G(F_2) := Surj(F_2,G)$$ which admits a natural action of $Aut(F_2)$. Furthermore, any surjection $p : G\twoheadrightarrow G'$ induces a natural map $$p_* : \underbrace{Surj(F_2,G)}_{G(F_2)}\longrightarrow \underbrace{Surj(F_2,G')}_{G'(F_2)}$$ which by Gaschutz' lemma is a surjection of $Aut(F_2)$-sets.

I wish to understand if:

- Every $Aut(F_2)$-equivariant automorphism $G(F_2)\rightarrow G(F_2)$ actually comes from an automorphism $\alpha\in Aut(G)$.
- Every $Aut(F_2)$-equivariant surjection $G(F_2)\rightarrow G'(F_2)$ actually comes from a surjection $G\rightarrow G'$.
- Same as (1), but now looking at $Aut(F_2)$-orbits. Ie, can you have an $Aut(F_2)$-equivariant automorphism of an $Aut(F_2)$-orbit of $G(F_2)$ that doesn't come from an automorphism $\alpha\in Aut(G)$?
- Same as (2), but looking at $Aut(F_2)$-orbits. Ie, can you have an $Aut(F_2)$-equivariant surjection from an $Aut(F_2)$-orbit of $G(F_2)$ to an orbit of $G'(F_2)$ that doesn't come from a surjection $G\rightarrow G'$?
- Are (1),(2),(3), or (4) true if we restrict the groups $G$ we consider? For example, if we restrict $G$ to only nonabelian finite simple groups (possibly letting $G'$ be anything)?

I mentioned the Yoneda embedding in the title because philosophically, I am asking if $G(F_2)$, as an $Aut(F_2)$-set, is "equivalent" to the functor $Surj(*,G)$.

Question (1) is probably the simplest, and I believe it is equivalent to asking:

(1a) Does the action of $Aut(F_2)$ on $\Omega_G := G(F_2)/Aut(G)$ have trivial centralizer inside the symmetric group on $\Omega_G$?

If $G$ is a nonabelian finite simple group, let $N_G := |\Omega_G|$. Then $N_G$ is called the $G$-rank of $F_2$, and in fact by a Goursat lemma argument, the product of the $N_G$ distinct representatives of $\Omega_G$ yields a surjection $F_2\twoheadrightarrow G^{N_G}$ with characteristic kernel. From this, if we fix an embedding $F_2\rightarrow\widehat{F_2}$ into the profinite completion, Gaschutz' lemma tells us that $Aut(\widehat{F_2})$ acts as the full symmetric group $S_{\Omega_G}$ on $\Omega_G$, which of course has trivial centralizer in $S_{\Omega_G}$. Of course, $Aut(F_2)$ is much much smaller than $Aut(\widehat{F_2})$, but on the other hand (1a) only requires that the centralizer be trivial.

Has this question been considered? I would also appreciate any relevant references.

Nielsen equivalence. $\endgroup$ – YCor Nov 1 '16 at 5:10